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arXiv:1011.3280 · Europhysics Letters (EPL)

Exact solutions to the Jaynes-Cummings model without the rotating-wave approximation

Qing‐Hu Chen, Tao Liu, Yuyu Zhang, Ke-Lin Wang

Published Sep 16, 2011• 44 citations
arXivHTMLPDF

Abstract

By using tunable extended bosonic coherent states, the JC model without the rotating-wave approximation can be mapped to a polynomial equation with a single variable. The solutions to this polynomial equation recover exactly all eigenvalues and eigenfunctions of the model for all coupling strengths and detunings, which can be readily applied to recent circuit quantum electrodynamic systems operating in the ultra-strong coupling regime.

DOI
10.1209/0295-5075/96/14003

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By using tunable extended bosonic coherent states, the JC model without the rotating-wave approximation can be mapped to a polynomial equation with a single variable. The solutions to this polynomial equation recover exactly all eigenvalues and eigenfunctions of the model for all coupling strengths and detunings, which can be readily applied to recent circuit quantum electrodynamic systems operating in the ultra-strong coupling regime.
This report presents an approach to the exact solutions of the time-dependent Jaynes-Cummings (J-C) model without the rotating wave approximation (RWA). It is shown that there is a squeezing-operator unitary transformation for relating the J-C model without RWA and the one with RWA. Thus by using an appropriate squeezing unitary transformation, the time-dependent J-C model without RWA can be transformed into the one with RWA that has been exactly solved previously, and based on this one can readily obtain the exact solutions of the time-dependent Jaynes-Cummings (J-C) model without RWA. The approach presented here also shows that we can treat these two kinds of time-dependent J-C models (both with and without RWA) in a unified way.
In this paper, we present a structure for obtaining the exact eigenfunctions and eigenvalues of the Jaynes–Cummings model (JCM) without the rotating wave approximation (RWA). We study the evolution of the system in the strong coupling region using the time evolution operator without RWA. The entanglement of the system without RWA is investigated using the Von Neumann entropy as an entanglement measure. It is interesting that in the weak coupling regime, the population of the atomic levels and Von Neumann entropy without RWA model shows a good agreement with the RWA whereas in strong coupling domain, the results of these two models are quite different.
The Jaynes-Cummings model without the rotating-wave approximation can be solved exactly by extended Swain's ansatz with the conserved parity. The analytical approximations are then performed at different levels. The well-known rotating-wave approximation is naturally covered in the present zero and first approximations. The effect of the counter rotating-wave term emerges clearly in the second order approximation. The concise analytical expressions are given explicitly and can be applicable up to the ultra-strong coupling regime. The preliminary application to the vacuum Rabi splitting is shown to be very successful.